Tài liệu hướng dẫn giải bài tập đại số tuyến tính

Exercises and Problems in Linear Algebra Exercises and Problems in Linear Algebra John M Erdman Portland State University Version July 13, 2014 c©2010 John M Erdman E mail address erdman@pdx edu Contents PREFACE vii Part 1 MATRICES AND LINEAR EQUATIONS 1 Chapter 1 SYSTEMS OF LINEAR EQUATIONS 3 1 1 Background 3 1 2 Exercises 4 1 3 Problems 7 1 4 Answers to Odd Numbered Exercises 8 Chapter 2 ARITHMETIC OF MATRICES 9 2 1 Background 9 2 2 Exercises 10 2 3 Problems 12 2 4 Answers to Odd Numbered Exer[.]

Exercises and Problems in Linear Algebra

John M Erdman Portland State University

Version July 13, 2014

c

E-mail address: erdman@pdx.edu

iii

iv CONTENTS

Chapter 15 SOME APPLICATIONS OF THE SPECTRAL THEOREM 97

Chapter 16 EVERY OPERATOR IS DIAGONALIZABLE PLUS NILPOTENT 105

CONTENTS v

vi CONTENTS

Chapter 26 SPECTRAL THEOREM FOR REAL INNER PRODUCT SPACES 171

Chapter 27 SPECTRAL THEOREM FOR COMPLEX INNER PRODUCT SPACES 177

This collection of exercises is designed to provide a framework for discussion in a junior levellinear algebra class such as the one I have conducted fairly regularly at Portland State University.There is no assigned text Students are free to choose their own sources of information Stu-dents are encouraged to find books, papers, and web sites whose writing style they find congenial,whose emphasis matches their interests, and whose price fits their budgets The short introduc-tory background section in these exercises, which precede each assignment, are intended only to fixnotation and provide “official” definitions and statements of important theorems for the exercisesand problems which follow

There are a number of excellent online texts which are available free of charge Among the bestare Linear Algebra [7] by Jim Hefferon,

to learn the subject, give them a look when you have the chance Another excellent traditionaltext is Linear Algebra: An Introductory Approach [5] by Charles W Curits And for those moreinterested in applications both Elementary Linear Algebra: Applications Version [1] by HowardAnton and Chris Rorres and Linear Algebra and its Applications [10] by Gilbert Strang are loadedwith applications

If you are a student and find the level at which many of the current beginning linear algebratexts are written depressingly pedestrian and the endless routine computations irritating, you mightexamine some of the more advanced texts Two excellent ones are Steven Roman’s Advanced LinearAlgebra [9] and William C Brown’s A Second Course in Linear Algebra [4]

Concerning the material in these notes, I make no claims of originality While I have dreamed

up many of the items included here, there are many others which are standard linear algebraexercises that can be traced back, in one form or another, through generations of linear algebratexts, making any serious attempt at proper attribution quite futile If anyone feels slighted, pleasecontact me

There will surely be errors I will be delighted to receive corrections, suggestions, or criticismat

vii

viii PREFACE

erdman@pdx.edu

I have placed the the LATEX source files on my web page so that those who wish to use these cises for homework assignments, examinations, or any other noncommercial purpose can downloadthe material and, without having to retype everything, edit it and supplement it as they wish

exer-Part 1

MATRICES AND LINEAR EQUATIONS

CHAPTER 1

SYSTEMS OF LINEAR EQUATIONS

1.1 BackgroundTopics: systems of linear equations; Gaussian elimination (Gauss’ method), elementary row op-erations, leading variables, free variables, echelon form, matrix, augmented matrix, Gauss-Jordanreduction, reduced echelon form

1.1.1 Definition We will say that an operation (sometimes called scaling) which multiplies a row

of a matrix (or an equation) by a nonzero constant is a row operation of type I An operation(sometimes called swapping) that interchanges two rows of a matrix (or two equations) is a rowoperation of type II And an operation (sometimes called pivoting ) that adds a multiple of onerow of a matrix to another row (or adds a multiple of one equation to another) is a row operation

of type III

3

4 1 SYSTEMS OF LINEAR EQUATIONS

1.2 Exercises(1) Suppose that L1 and L2 are lines in the plane, that the x-intercepts of L1 and L2 are 5and −1, respectively, and that the respective y-intercepts are 5 and 1 Then L1 and L2intersect at the point ( , )

(2) Consider the following system of equations

(a) List the leading variables

(b) List the free variables

(c) The general solution of (∗) (expressed in terms of the free variables) is

(3) Consider the following system of equations:

(c) The solutions of (∗) are x = , y = , and z =

(4) Consider the following system of equations

0.003000x + 59.14y = 59.175.291x − 6.130y = 46.78

(a) Using only row operation III and back substitution find the exact solution of thesystem Answer: x = , y =

(b) Same as (a), but after performing each arithmetic operation round off your answer tofour significant figures Answer: x = , y =

1.2 EXERCISES 5

(5) Find the values of k for which the system of equations

x + ky = 1

kx + y = 1

(c) infinitely many solutions Answer:

(d) When there is exactly one solution, it is x = and y =

(6) Consider the following two systems of equations

(c) The solution for (1) is ( , , ) and the solution for (2) is ( , , )

(7) Consider the following system of equations:

(b) For the value of c you found in (a) describe the solution set geometrically as a subset

(c) What does part (a) say about the planes x − y − 3z = 3, 2x + z = 0, and 2y + 7z = 4

6 1 SYSTEMS OF LINEAR EQUATIONS

(8) Consider the following system of linear equations ( where b1, , b5 are constants)

(a) In the process of Gaussian elimination the leading variables of this system are

and the free variables are

(b) What condition(s) must the constants b1, , b5 satisfy so that the system is

(b) Suppose that a does not have the value you found in part (a) Find a value for b sothat the system has a nontrivial solution

Answer: b = c3+ d3a where c = and d =

(c) Suppose that a does not have the value you found in part (a) and that b = 100.Suppose further that a is chosen so that the solution to the system is not unique.The general solution to the system (in terms of the free variable) is α1 z , −β1z , z

1.3 PROBLEMS 7

1.3 Problems(1) Give a geometric description of a single linear equation in three variables

Then give a geometric description of the solution set of a system of 3 linear equations in

3 variables if the system

(a) is inconsistent

(b) is consistent and has no free variables

(c) is consistent and has exactly one free variable

(d) is consistent and has two free variables

(2) Consider the following system of equations:

−m1x + y = b1

−m2x + y = b2

(∗)(a) Prove that if m1 6= m2, then (∗) has exactly one solution What is it?

(b) Suppose that m1 = m2 Then under what conditions will (∗) be consistent?

(c) Restate the results of (a) and (b) in geometrical language

8 1 SYSTEMS OF LINEAR EQUATIONS

1.4 Answers to Odd-Numbered Exercises(1) 2, 3

CHAPTER 2

ARITHMETIC OF MATRICES

2.1 BackgroundTopics: addition, scalar multiplication, and multiplication of matrices, inverse of a nonsingularmatrix

2.1.1 Definition Two square matrices A and B of the same size are said to commute if AB =BA

2.1.2 Definition If A and B are square matrices of the same size, then the commutator (orLie bracket) of A and B, denoted by [A, B], is defined by

[A, B] = AB − BA 2.1.3 Notation If A is an m × n matrix (that is, a matrix with m rows and n columns), then theelement in the ith row and the jth column is denoted by aij The matrix A itself may be denoted

2.1.4 Definition A matrix A = [aij] is upper triangular if aij = 0 whenever i > j

2.1.5 Definition The trace of a square matrix A, denoted by tr A, is the sum of the diagonalentries of the matrix That is, if A = [aij] is an n × n matrix, then

2.1.6 Definition The transpose of an n × n matrix A =aij is the matrix At=aji obtained

by interchanging the rows and columns of A The matrix A is symmetric if At= A

2.1.7 Proposition If A is an m × n matrix and B is an n × p matrix, then (AB)t= BtAt

9

(a) Does the matrix D = ABC exist? If so, then d34 =

(b) Does the matrix E = BAC exist? If so, then e22 =

(c) Does the matrix F = BCA exist? If so, then f43 =

(d) Does the matrix G = ACB exist? If so, then g31=

(e) Does the matrix H = CAB exist? If so, then h21 =

(f) Does the matrix J = CBA exist? If so, then j13 =

(2) Let A =

"1 2

1 2 1 2

1 2

#, B =1 0

0 −1

, and C = AB Evaluate the following

where a is a real number

(a) For what value of a will a row interchange be required during Gaussian elimination?Answer: a =

(b) For what value of a is the matrix singular? Answer: a =

, and

(9) If A is an n × n matrix and it satisfies the equation A3− 4A2+ 3A − 5In= 0, then A isnonsingular

2.2 EXERCISES 11

(10) Let A, B, and C be n × n matrices Then [[A, B], C] + [[B, C], A] + [[C, A], B] = X, where

(11) Let A, B, and C be n × n matrices Then [A, C] + [B, C] = [X, Y ], where X =

1

3 1 0

1 2

1 2

1 3

1 4

1 5 1

3

1 4

1 5

1 6 1

4

1 5

1 6

1 7

is the 4 × 4 Hilbert matrix Use Gauss-Jordan elimination to compute K = H−1 Then

K44 is (exactly) Now, create a new matrix H0 by replacing each entry in H

by its approximation to 3 decimal places (For example, replace 16 by 0.167.) Use Jordan elimination again to find the inverse K0 of H0 Then K440 is (14) Suppose that A and B are symmetric n × n matrices In this exercise we prove that AB

Gauss-is symmetric if and only if A commutes with B Below are portions of the proof Fill inthe missing steps and the missing reasons Choose reasons from the following list

(H1) Hypothesis that A and B are symmetric

(H2) Hypothesis that AB is symmetric

(H3) Hypothesis that A commutes with B

(D1) Definition of commutes

(D2) Definition of symmetric

(T) Proposition2.1.7.Proof Suppose that AB is symmetric Then

= BtAt (reason: )

So A commutes with B (reason: )

Conversely, suppose that A commutes with B Then

12 2 ARITHMETIC OF MATRICES

2.3 Problems(1) Let A be a square matrix Prove that if A2 is invertible, then so is A

Hint Our assumption is that there exists a matrix B such that

A2B = BA2= I

We want to show that there exists a matrix C such that

AC = CA = I Now to start with, you ought to find it fairly easy to show that there are matrices L and

R such that

A matrix L is a left inverse of the matrix A if LA = I; and R is a right inverse

of A if AR = I Thus the problem boils down to determining whether A can have a leftinverse and a right inverse that are different (Clearly, if it turns out that they must bethe same, then the C we are seeking is their common value.) So try to prove that if (∗)holds, then L = R

(2) Anton speaks French and German; Geraldine speaks English, French and Italian; Jamesspeaks English, Italian, and Spanish; Lauren speaks all the languages the others speakexcept French; and no one speaks any other language Make a matrix A = aij withrows representing the four people mentioned and columns representing the languages theyspeak Put aij = 1 if person i speaks language j and aij = 0 otherwise Explain thesignificance of the matrices AAt and AtA

(3) Portland Fast Foods (PFF), which produces 138 food products all made from 87 basicingredients, wants to set up a simple data structure from which they can quickly extractanswers to the following questions:

(a) How many ingredients does a given product contain?

(b) A given pair of ingredients are used together in how many products?

(c) How many ingredients do two given products have in common?

(d) In how many products is a given ingredient used?

In particular, PFF wants to set up a single table in such a way that:

(i) the answer to any of the above questions can be extracted easily and quickly (matrixarithmetic permitted, of course); and

(ii) if one of the 87 ingredients is added to or deleted from a product, only a single entry

in the table needs to be changed

Is this possible? Explain

(4) Prove proposition 2.1.7

(5) Let A and B be 2 × 2 matrices

(a) Prove that if the trace of A is 0, then A2 is a scalar multiple of the identity matrix.(b) Prove that the square of the commutator of A and B commutes with every 2 × 2matrix C Hint What can you say about the trace of [A, B]?

(c) Prove that the commutator of A and B can never be a nonzero multiple of the identitymatrix

2.3 PROBLEMS 13

(6) The matrices that represent rotations of the xy-plane are

A(θ) =cos θ − sin θ

sin θ cos θ



(a) Let x be the vector (−1, 1), θ = 3π/4, and y be A(θ) acting on x (that is, y = A(θ)xt).Make a sketch showing x, y, and θ

(b) Verify that A(θ1)A(θ2) = A(θ1+ θ2) Discuss what this means geometrically

(c) What is the product of A(θ) times A(−θ)? Discuss what this means geometrically.(d) Two sheets of graph paper are attached at the origin and rotated in such a way thatthe point (1, 0) on the upper sheet lies directly over the point (−5/13, 12/13) on thelower sheet What point on the lower sheet lies directly below (6, 4) on the upperone?

The goal of this problem is to develop a “calculus” for the matrix A To start, recall(or look up) the power series expansion for 1

1 − x Now see if this formula works forthe matrix A by first computing (I − A)−1 directly and then computing the power seriesexpansion substituting A for x (Explain why there are no convergence difficulties for theseries when we use this particular matrix A.) Next try to define ln(I + A) and eA bymeans of appropriate series Do you get what you expect when you compute eln(I+A)? Doformulas like eAeA = e2A hold? What about other familiar properties of the exponentialand logarithmic functions?

Try some trigonometry with A Use series to define sin, cos, tan, arctan, and so on Dothings like tan(arctan(A)) produce the expected results? Check some of the more obvioustrigonometric identities (What do you get for sin2A + cos2A − I? Is cos(2A) the same

as cos2A − sin2A?)

A relationship between the exponential and trigonometric functions is given by thefamous formula eix = cos x + i sin x Does this hold for A?

Do you think there are other matrices for which the same results might hold? Whichones?

(8) (a) Give an example of two symmetric matrices whose product is not symmetric

Hint Matrices containing only 0’s and 1’s will suffice

(b) Now suppose that A and B are symmetric n×n matrices Prove that AB is symmetric

if and only if A commutes with B

Hint To prove that a statement P holds “if and only if” a statement Q holds you mustfirst show that P implies Q and then show that Q implies P In the current problem, thereare 4 conditions to be considered:

(i) At= A (A is symmetric),

(ii) Bt= B (B is symmetric),

(iii) (AB)t= AB (AB is symmetric), and

(iv) AB = BA (A commutes with B)

Recall also the fact given in

(v) theorem 2.1.7

The first task is to derive (iv) from (i), (ii), (iii), and (v) Then try to derive (iii) from (i),(ii), (iv), and (v)

CHAPTER 3

ELEMENTARY MATRICES; DETERMINANTS

3.1 BackgroundTopics: elementary (reduction) matrices, determinants

The following definition says that we often regard the effect of multiplying a matrix M on theleft by another matrix A as the action of A on M

3.1.1 Definition We say that the matrix A acts on the matrix M to produce the matrix N if

N = AM For example the matrix 0 1

1 0

acts on any 2 × 2 matrix by interchanging (swapping)its rows because0 1

3.1.2 Notation We adopt the following notation for elementary matrices which implement type Irow operations Let A be a matrix having n rows For any real number r 6= 0 denote by Mj(r) the

n × n matrix which acts on A by multiplying its jth row by r (See exercise 1.)

3.1.3 Notation We use the following notation for elementary matrices which implement type IIrow operations (See definition1.1.1.) Let A be a matrix having n rows Denote by Pij the n × nmatrix which acts on A by interchanging its ith and jth rows (See exercise2.)

3.1.4 Notation And we use the following notation for elementary matrices which implementtype III row operations (See definition 1.1.1.) Let A be a matrix having n rows For any realnumber r denote by Eij(r) the n × n matrix which acts on A by adding r times the jth row of A

to the ith row (See exercise 3.)

3.1.5 Definition If a matrix B can be produced from a matrix A by a sequence of elementaryrow operations, then A and B are row equivalent

Some Facts about Determinants3.1.6 Proposition Let n ∈ N and Mn×n be the collection of all n × n matrices There is exactlyone function

det : Mn×n→ R : A 7→ det Awhich satisfies

(d) If A ∈ Mn×n, c ∈ R, and A0 is the matrix obtained from A by multiplying one row of A

by c and adding it to another row of A (that is, choose i and j between 1 and n with i 6= jand replace ajk by ajk+ caik for 1 ≤ k ≤ n), then det A0 = det A

15

16 3 ELEMENTARY MATRICES; DETERMINANTS

3.1.7 Definition The unique function det : Mn×n → R described above is the n × n nant function

determi-3.1.8 Proposition If A = [a] for a ∈ R (that is, if A ∈ M1×1), then det A = a; if A ∈ M2×2,then

det A = a11a22− a12a21.3.1.9 Proposition If A, B ∈ Mn×n, then det(AB) = (det A)(det B)

3.1.10 Proposition If A ∈ Mn×n, then det At = det A (An obvious corollary of this: inconditions (b), (c), and (d) of proposition 3.1.6 the word “columns” may be substituted for theword “rows”.)

3.1.11 Definition Let A be an n × n matrix The minor of the element ajk, denoted by Mjk, isthe determinant of the (n − 1) × (n − 1) matrix which results from the deletion of the jth row and

kth column of A The cofactor of the element ajk, denoted by Cjk is defined by

Cjk := (−1)j+kMjk.3.1.12 Proposition If A ∈ Mn×n and 1 ≤ j ≤ n, then

This is the (Laplace) expansion of the determinant along the jth row

In light of 3.1.10, it is clear that expansion along columns works as well as expansion alongrows That is,

3.2 EXERCISES 17

3.2 Exercises(1) Let A be a matrix with 4 rows The matrix M3(4) which multiplies the 3rd row of A by 4

(2) Let A be a matrix with 4 rows The matrix P24 which interchanges the 2nd and 4th rows

(3) Let A be a matrix with 4 rows The matrix E23(−2) which adds −2 times the 3rd row of

(4) Let A be the 4 × 4 elementary matrix E43(−6) Then A11=

(5) Let B be the elementary 4 × 4 matrix P24 Then B−9 =

(6) Let C be the elementary 4 × 4 matrix M3(−2) Then C4 =

and b32 =

(8) We apply Gaussian elimination (using type III elementary row operations only) to put a

4 × 4 matrix A into upper triangular form The result is

E43(52)E42(2)E31(1)E21(−2)A =

Then the determinant of A is

18 3 ELEMENTARY MATRICES; DETERMINANTS

(9) The system of equations:







2y+3z = 7x+ y− z = −2

 Give the names of the elementary 3 × 3 matrices X1, , X8

which implement the following reduction

 Find A−1using the technique of augmenting A by the identity matrix

I and performing Gauss-Jordan reduction on the augmented matrix The reduction can

be accomplished by the application of five elementary 3 × 3 matrices Find elementarymatrices X1, X2, and X3 such that A−1= X3E13(−3)X2M2(1/2)X1I

(a) The required matrices are X1= P1iwhere i = , X2= Ejk(−2) where j =and k = , and X3 = E12(r) where r =

(a) The determinant of M can be expressed as the constant 5 times the determinant ofthe single 3 × 3 matrix





33

(d) Thus the determinant of M is

(15) Find the determinant of the matrix

Hint Do not use a calculator Answer:

20 3 ELEMENTARY MATRICES; DETERMINANTS

22 3 ELEMENTARY MATRICES; DETERMINANTS

3.3 Problems(1) For this problem assume that we know the following: If X is an m × m matrix, if Y is

an m × n matrix and if 0 and I are zero and identity matrices of appropriate sizes, thendetX Y

(b) Is the result he is trying to prove actually true?

Hint: Consider the productI B

0 A − B

 A + B 0

.(3) Let x be a fixed real number which is not an integer multiple of π For each naturalnumber n let An=ajk be the n × n-matrix defined by

Show that det An= sin(n + 1)x

sin x Hint For each integer n let Dn= det Anand prove that

Dn+2− 2Dn+1cos x + Dn= 0

(Use mathematical induction.)

3.4 ANSWERS TO ODD-NUMBERED EXERCISES 23

3.4 Answers to Odd-Numbered Exercises

CHAPTER 4

4.1 BackgroundTopics: inner (dot) products, cross products, lines and planes in 3-space, norm of a vector, anglebetween vectors

4.1.1 Notation There are many more or less standard notations for the inner product (or dotproduct) of two vectors x and y The two that we will use interchangeably in these exercises are

x · y and hx, yi

4.1.2 Definition If x is a vector in Rn, then the norm (or length) of x is defined by

kxk =phx, xi 4.1.3 Definition Let x and y be nonzero vectors in Rn Then ](x, y), the angle between xand y, is defined by

](x, y) = arccoskxk kykhx, yi4.1.4 Theorem (Cauchy-Schwarz inequality) If x and y are vectors in Rn, then

|hx, yi| ≤ kxk kyk (We will often refer to this just as the Schwarz inequality.)

4.1.5 Definition If x = (x1, x2, x3) and y = (y1, y2, y3) are vectors in R3, then their crossproduct, denoted by x × y, is the vector (x2y3− x3y2, x3y1− x1y3, x1y2− x2y1)

25

26 4 VECTOR GEOMETRY IN R

4.2 Exercises(1) The angle between the vectors (1, 0, −1, 3) and (1,√3, 3, −3) in R4is aπ where a = (2) Find the angle θ between the vectors x = (3, −1, 1, 0, 2, 1) and y = (2, −1, 0,√2, 2, 1)

(5) Which of the angles (if any) of triangle ABC, with A = (1, −2, 0), B = (2, 1, −2), and

C = (6, −1, −3), is a right angle? Answer: the angle at vertex

(6) The hydrogen atoms of a methane molecule (CH4) are located at (0, 0, 0), (1, 1, 0), (0, 1, 1),and (1, 0, 1) while the carbon atom is at (12,12,12) Find the cosine of the angle θ betweentwo rays starting at the carbon atom and going to different hydrogen atoms

Answer: cos θ =

(7) If a, b, c, d, e, f ∈ R, then

|ad + be + cf | ≤pa2+ b2+ c2pd2+ e2+ f2.The proof of this inequality is obvious since this is just the Cauchy-Schwarz inequality

4.2 EXERCISES 27

(15) Suppose that u ∈ R3 is a vector which lies in the first quadrant of the xy-plane and haslength 3 and that v ∈ R3 is a vector that lies along the positive z-axis and has length 5.Then

(a) ku × vk = ;

(b) the x-coordinate of u × v is 0 (choose <, >, or =);

(c) the y-coordinate of u × v is 0 (choose <, >, or =); and

(d) the z-coordinate of u × v is 0 (choose <, >, or =)

(16) Suppose that u and v are vectors in R7 both of length 2√2 and that the length of u − v

is also 2√2 Then ku + vk = and the angle between u and v is

28 4 VECTOR GEOMETRY IN R

4.3 Problems(1) Show that if a, b, c > 0, then 12a +13b + 16c
2≤ 12a2+13b2+16c2

(2) Show that if a1, , an, w1, , wn> 0 and Pn

You may find the following steps helpful in organizing your solution

(i) First of all, make sure that you recall the difference between a sequence of numbers(c1, c2, ) and an infinite series

∞

X

k=1

ck

(ii) The key to this problem is an important theorem from third term Calculus:

A nondecreasing sequence of real numbers converges if and only if it is bounded (∗)(Make sure that you know the meanings of all the terms used here.)

(iii) The hypothesis of the result we are trying to prove is that the series

∞

X

k=1

ak2 converges.What, exactly, does this mean?

(iv) For each natural number n let bn =

(v) Is the sequence (bn) nondecreasing?

(vi) What, then, does (∗) say about the sequence (bn)?

(vii) For each natural number n let cn =

(ix) For each natural number n let sn=

4.4 ANSWERS TO ODD-NUMBERED EXERCISES 29

4.4 Answers to Odd-Numbered Exercises(1) 3

Part 2

VECTOR SPACES